cp's OEIS Frontend

A097792 Numbers of the form 4k^4 or (kp)^p for prime p > 2 and k = 1, 2, 3, ....

%I A097792 #31 Jul 08 2024 11:13:15
%S A097792 4,27,64,216,324,729,1024,1728,2500,3125,3375,5184,5832,9261,9604,
%T A097792 13824,16384,19683,26244,27000,35937,40000,46656,58564,59319,74088,
%U A097792 82944,91125,100000,110592,114244,132651,153664,157464,185193,202500,216000
%N A097792 Numbers of the form 4k^4 or (kp)^p for prime p > 2 and k = 1, 2, 3, ....
%C A097792 A result of Vahlen shows that the polynomial x^n + n is reducible over the integers for n in this sequence and no other n.
%H A097792 David A. Corneth, <a href="/A097792/b097792.txt">Table of n, a(n) for n = 1..10000</a>
%H A097792 A. Schinzel, <a href="http://algo.inria.fr/seminars/sem92-93/schinzel.pdf">Problems and results on polynomials</a>, Algorithms Seminar, INRIA, 1992-1993.
%H A097792 K. T. Vahlen, <a href="http://dx.doi.org/10.1007/BF02402875">Über reductible Binome</a>, Acta Mathematica 19:1 (December 1895), pp. 195-198.
%F A097792 Is a(n) ~ c * n^3? - _David A. Corneth_, Jan 12 2019
%t A097792 nMax=500000; lst={}; k=1; While[4k^4<=nMax, AppendTo[lst, 4k^4]; k++ ]; n=2; While[p=Prime[n]; p^p<=nMax, k=1; While[(k*p)^p<=nMax, AppendTo[lst, (k*p)^p]; k++ ]; n++ ]; Union[lst]
%o A097792 (PARI) upto(n) = {my(res = List()); for(i = 1, sqrtnint(n \ 4, 4), listput(res, 4*i^4) ); forprime(p = 3, log(n), pp = p^p; for(k = 1, sqrtnint(n \ pp, p), listput(res, pp * k ^ p); ) ); listsort(res); res } \\ _David A. Corneth_, Jan 12 2019
%o A097792 (PARI) select( {is_A097792(n, p=0)= n%4==0 && ispower(n\4,4) || ((2 < p = ispower(n,,&n)) && if(isprime(p), n%p==0, foreach(factor(p)[,1], q, q%2 && n%q==0 && return(1))))}, [1..10^4]) \\ _M. F. Hasler_, Jul 07 2024
%o A097792 (Python)
%o A097792 from sympy import isprime, perfect_power, primefactors
%o A097792 def is_A097792(n):
%o A097792     return n%4==0 and (perfect_power(n//4,[4]) or n==4) or (
%o A097792         p := perfect_power(n)) and p[1] > 2 and (p[0]%p[1]==0 if isprime(p[1])
%o A097792         else any(p[0]%q==0 for q in primefactors(p[1]) if q > 2))
%o A097792 # _M. F. Hasler_, Jul 07 2024
%Y A097792 Cf. A093324 (least k such that n^k+k is prime), A097764 (numbers of the form (kp)^p).
%Y A097792 Cf. A072883, A239666, A303121, A303122.
%K A097792 nonn
%O A097792 1,1
%A A097792 _T. D. Noe_, Aug 24 2004